Numerical studies on the zakharov system

17 3 Numerical Methods for the Zakharov System 19 3.1 Time-splitting spectral discretizations TSSP.. Furthermore, Conservation laws ofthe system are proven, relation to the nonlinear Sch

NUMERICAL STUDIES ON THE

ZAKHAROV SYSTEM

SUN FANGFANG

(B.Sc., Jilin University)

A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF SCIENCE DEPARTMENT OF COMPUTATIONAL SCIENCE NATIONAL UNIVERSITY OF SINGAPORE

2003

I would like to thank my supervisor, Dr Bao Weizhu, who gave me the nity to work on such an interesting research project, paid patient guidance to me,reviewed my thesis and gave me much invaluable help and constructive suggestion

opportu-on it

It is also my pleasure to express my appreciation and gratitude to A/P WeiGuowei, from whom I got effective training on programming, good ideas and expe-rience, both of which are the foundation for my subsequent research project, andmuch valuable suggestion on my research project

I would also wish to thank the National University of Singapore for her financialsupport by awarding me the Research Scholarship during the period of my MSccandidature

My sincere thanks go to all my department-mates for their friendship and so muchkind help And special thanks go to Mr Zhao Shan for his patient help whenever Iencountered problems in my research

ii

Acknowledgments iii

I would like also to dedicate this work to my parents, who love me most in theworld, for their unconditional love and support

Sun FangfangJune 2003

1.1 Physical background 1

1.2 The problem 2

1.3 Review of existing results 3

1.4 Our main results 3

2 The Zakharov system 6 2.1 Derivation of the vector Zakharov system 6

2.2 Simplification and generalization 11

iv

Contents v

2.3 Relation to the nonlinear Schr¨odinger equation(NLS) 13

2.4 Conservation laws of the system 13

2.5 Well-posedness of the Zakharov system (ZS) 16

2.6 Plane wave and soliton wave solutions 17

3 Numerical Methods for the Zakharov System 19 3.1 Time-splitting spectral discretizations (TSSP) 20

3.1.1 The numerical method 20

3.1.2 For plane wave solution 23

3.1.3 Conservation and decay rate 25

3.2 Other numerical methods 28

3.2.1 Discrete singular convolution (DSC-RK4) 28

3.2.2 Fourier pseudospectral method (FPS-RK4) 31

3.2.3 Wavelet-Galerkin method (WG-RK4) 32

3.2.4 Finite difference method (FD) 33

3.3 Extension TSSP to Zakharov system for multi-component plasma 34

3.4 Extension TSSP to vector Zakharov system 35

4 Numerical Examples 38 4.1 Comparisons of different methods 38

4.2 Applications of TSSP 45

4.2.1 Plane-wave solution of the standard Zakharov system 45

4.2.2 Soliton-soliton collisions of the standard Zakharov system 46

4.2.3 Solution of 2d standard Zakharov system 50

4.2.4 Soliton-soliton collisions of the generalized Zakharov system 52 4.2.5 Solutions of the damped Zakharov system 56

Contents vi

In this thesis, we present two numerical methods for studying solutions of the kharov system (ZS) We begin with the vector ZS derived from the two-fluid model,and simplify the vector ZS to get the standard ZS, then extend it for multicompo-nent plasma and finally get the generalized ZS Furthermore, Conservation laws ofthe system are proven, relation to the nonlinear Schr¨odinger equation (NLS), planewave and soliton wave solutions, as well as well-posedness of the ZS are reviewed.Then we proposed two numerical methods for the approximation of the generalizedZakharov system The first one is the time-splitting spectral (TSSP) method, which

Za-is explicit, keeps the same decay rate of a standard variant as that in the alized ZS, gives exact results for the plane-wave solution, and is of spectral-orderaccuracy in space and second-order accuracy in time The second one is to usethe discrete singular convolution (DSC) for spatial derivatives and the fourth-orderRunge-Kutta (RK4) for time integration, which is of high (the same as spectral) or-der accuracy in space and can be applied to deal with general boundary conditions.Furthermore, extension of TSSP to the vector ZS as well as ZS for multi-componentplasma are presented In order to test accuracy and stability, we compare these twomethods with other existing methods: Fourier pseudospectral method (FPS) andwavelet-Galerkin method (WG) for spatial derivatives combining with RK4 for time

gener-vii

Summary viii

integration, as well as the standard finite difference method (FD) for solving the ZSwith a solitary-wave solution Furthermore, extensive numerical tests are presentedfor plane waves, colliding solitary waves in 1d, a 2d problem as well as a dampedproblem of a generalized ZS

The thesis is organized as follows: In Chapter 1, the physical background of theZakharov system is introduced, and we review some existing results and report ourmain results In Chapter 2, the Zakharov system are derived and their properties areanalyzed Chapter 3 is devoted to present the time-splitting spectral discretizationand DSC algorithm of the generalized Zakharov system In Chapter 4, we comparethe accuracy and stability of different methods for the ZS with a solitary wavesolution, as well as present numerical results for plane waves, soliton-soliton collisions

in 1d, 2d problem, the generalized ZS with a damping term and ZS for component plasma Finally, some conclusions based on our findings and numericalresults are drawn in Chapter 5

multi-List of Symbols

PARAMETERS AND VARIABLES

ix

List of Symbols x

ˆ

List of Symbols xi

List of Tables

0.00001 40

4.2 Time discretization error analysis: e1, e2 at time t=2 41

4.3 Conserved quantities analysis: k = 0.001 and h = 18 42

4.4 Stability analysis: time t = 5.0 43

4.5 Parameter values for analytic solutions of the periodic Zakharov sys-tem 46

xii

List of Figures

8,

h = 12, k = 501 ; b) ε = 321, h = 18, k = 8001 ; c) ε = 1281 , h = 321,

12800 corresponding to h = O(ε) and k = O(εh) = O(ε2) 44

‘—’: exact solution given in (4.6)-(4.7), ‘+ + +’: numerical solution.a) Re(E(x, t)): real part of E, b) Im(E(x, t)): imaginary part of E,c) N 47

field |E(x, t)| 49

xiii

List of Figures xiv

4.10 Numerical solutions in Example 5 for case 3 a) Evolution of the

4.11 Numerical solutions in Example 5 for case 4 a) Evolution of the

4.12 Numerical results in Example 6 for case 1 Surface-plot of the electric

times: a) t = 0, b) t = 0.5, c) t = 1.0 634.13 Numerical results in Example 6 for case 1 Surface-plot of the electric

times: a) Before blow up (t=0.7), b) After blow up (t=1.333) 644.14 Numerical results in Example 6 for case 1 Surface-plot of the electric

times: a) Before blow up (t=0.5), b) After blow up (t=1.177) 654.15 Numerical results in Example 6 for case 2 Surface-plot of the electric

times: a) t = 0, b) t = 0.5, c) t = 1.0 664.16 Numerical results in Example 6 for case 2 Surface-plot of the electric

times: a) Before blow up (t=0.2), b) After blow up (t=0.473) 674.17 Numerical results in Example 6 for case 2 Surface-plot of the electric

times: a) Before blow up (t=0.2), b) After blow up (t=0.442) 68

List of Figures xv

4.18 Numerical results in Example 6 for case 3 Surface-plot of the electric

times: a) t = 0, b) t = 0.5, c) t = 1.0 694.19 Numerical results in Example 6 for case 3 Surface-plot of the electric

4.20 Numerical results in Example 6 for case 3 Surface-plot of the electric

4.21 Numerical results in Example 6 for three cases: Energy, electric field

and ion density as functions of time with γ = 0.8 (left: no blow up)

, γ = 0.1 (center: blow up) and γ = 0 (right: blow up) a) Case 1,

b) Case 2, c) Case 3 72

Chapter 1

Introduction

Zakharov [52] derived the Zakharov system(ZS) governing the coupled dynamics

of the electric-field amplitude and of the low-frequency density fluctuations of theions Later, it has become commonly accepted that ZS is a general model to governinteraction of dispersive and nondispersive waves It has important applications

in plasma physics (interaction between Langmuir and ion acoustic waves [52]), inthe theory of molecular chains (interaction of the intramolecular vibrations formingDavydov solitons with the acoustic disturbances in the chain [17]), in hydrodynamics(interaction between short-wave and long-wave gravitational disturbances in theatmosphere [35]), and so on In three spatial dimensions, ZS was also derived tomodel the collapse of caverns [52] Since then a combined numerical and analyticalattack has been launched on ZS As is well known, ZS is not exactly integrable [37],

so the numerical solution is very important

1

is a parameter inversely proportional to the acoustic speed, γ ≥ 0 is a dampingparameter, and α, λ, ν are all real constants.

The general form of (1.1) and (1.2) covers many different generalized Zakharov tems arising in various physical applications For example: a) when ε = 1, ν = −1,

sys-λ = 0, γ = 0 and α = 1, the system of eqs (1.1) and (1.2) reduces to the well-knownZakharov system, which has been first derived by Zakharov [52] to describe theinteraction between Langmuir (dispersive) and ion acoustic (approximately nondis-persive) waves in a plasma Since then, it has become commonly accepted thatthe ZS is a general model governing the interaction of dispersive and nondispersivewaves; b) when ε = 1, ν = −1 and λ 6= 0, a cubic nonlinearity is added to thefirst equation (1.1); c) when γ > 0, a linear damping term is added to the ZS; d)

which, together with (1.1), leads to the well-known nonlinear Schr¨odinger equation(NLS) without (γ = 0) or with (γ > 0) a linear damping term:

1.3 Review of existing results 3

The global existence of a weak solution of the Zakharov equations in 1d is proven

in [40], and the existence and uniqueness of the smooth solution for the equationsare obtained under the ground that smooth initial data are prescribed The well-posedness of the ZS was recently improved in [10] for d = 1, 2, 3, and extended forthe case with generalized nonlinearity [15]

On the other hand, numerical methods for the standard Zakharov system, i.e ε = 1,

ν = −1, λ = 0, γ = 0 in (1.1) and (1.2), were studied in the last two decades Payne

et al [31] proposed a Fourier spectral method for the 1d Zakharov system Theyused only two-thirds of the Fourier components for a particular mesh in the fastFourier transform in order to suppress the aliasing errors in their algorithm [31]

Of course, this is not an optimal way to use spectral method In [19, 20], Glasseypresented an energy-preserving implicit finite difference scheme for the system andproved its convergence Later, Chang et al [12] considered an implicit or semiex-plicit conservative finite difference scheme for the ZS, proved its convergence, andextended their method for the generalized Zakharov system [13] More numericalstudy of soliton-soliton collisions for a (generalized) Zakharov system can be found

in [29, 24, 25]

In this thesis, we propose a time-splitting spectral (TSSP) approximation and adiscrete singular convolution (DSC) algorithm for the generalized Zakharov sys-tem TSSP is explicit, easy to extend to high dimension, and gives exact resultsfor plane-wave solutions of the ZS For stability, TSSP requires k = O(h) In fact,the spectral method has showed greatly success in solving problems arising frommany areas [4, 11, 21] and the split-step procedure was presented for differential

1.4 Our main results 4

equations [39] and applied to Schr¨odinger equation [18, 27, 42] and KDV equation[43] Recently, the time-splitting spectral approximation was studied and used fornonlinear Schr¨odinger equation in the semiclassical regimes in [7] and applied to thenumerical study of the dynamics of Bose-Einstein condensation [8] as well Verypromising numerical results were obtained due to its exponentially high order accu-racy in space The approach for the ZS is based on a time splitting for (1.1) which

Another method we will present for ZS is the discrete singular convolution(DSC)method which was recently proposed by Wei [45] as a potential approach for thenumerical discretization of spatial derivatives The main merit of the DSC method

is that it is of high (the same as spectral) order accuracy in space and can beapplied to deal with complex geometries and general boundary conditions So farthis method has been successfully applied to solve many problems in science andengineering, such as eigenvalue problems of both quantum [47] and classical [48]origins, analysis of stochastic process [45, 46], simulation of fluid flow in simple [49]and complex geometries, and nanoscale pattern formation in a circular domain [23].DSC method has the theory of distribution as its mathematical foundation [38].And numerical analysis indicates that the DSC method has spectral convergence forapproximating appropriate functions [3, 4] The accuracy and stability of TSSP andDSC will be compared with other existing methods like finite difference method.The numerical results demonstrate the high accuracy and efficiency of these twoproposed methods for the ZS

This thesis consists of four Chapters arranged as following Chapter 1 introducethe physical background of the Zakharov system, and we also review some existingresults and report our main results In Chapter 2, the Zakharov system, whichgoverns the coupled dynamics of the electric-field amplitude and of the low-frequencydensity fluctuations of the ions, is derived and its properties are analyzed Chapter 3

is devoted to present the time-splitting spectral discretization and DSC algorithm for

1.4 Our main results 5

the generalized Zakharov system and some other existing methods are introduced,too Furthermore, extension TSSP to ZS for multi-component and vector ZS InChapter 4, we will compare the accuracy and stability of different methods for the

ZS with a solitary wave solution, and also present the numerical results for planewaves, soliton-soliton collisions in 1d, 2d problems and the generalized ZS with adamping term Finally, some conclusions based on our findings and numerical resultsare drawn in Chapter 5

Chapter 2

The Zakharov system

In this Chapter, We firstly review the derivation of the vector ZS from the fluid model [41], and simplify the vector ZS to get the standard ZS, then extend

two-it in a multicomponent plasma and finally get the generalized ZS wtwo-ith a dampingterm Furthermore, Conservation laws of the system are proved and relation to thenonlinear Schr¨odinger equation(NLS), plane wave and soliton wave solutions, as well

as well-posedness of the ZS are reviewed

This section is devoted to derive the Zakharov system from the two-fluid model [41].Here we will use a more formal approach based on the multiple-scale modulationanalysis Following from [41], we will consider a plasma as two interpenetratingfluids, an electron fluid and an ion fluid, and denote the mass, number density(number of particles per unit volume) and velocity of the electrons (respectively of

continuity equations for these fluids read

6

2.1 Derivation of the vector Zakharov system 7and the momentum equations read

where −e and e represent the charge of the electron and the ions assumed to reduce

energy units The electric field E and magnetic field B are provided by the Maxwellequations

and total current, respectively

Equations (2.7) and (2.8) yield

In order to get the vector Zakharov system from the two-fluid model just mentioned,

as in [41], we consider a long-wavelength small-amplitude Langmuir oscillation ofthe form

2.1 Derivation of the vector Zakharov system 8

where the complex amplitude E depends on the slow variables X = εx and T =

mean complex amplitude It induces fluctuations for the density and velocity ofthe electrons and of the ions whose dynamical time will be seen to be τ = εt, thusshorter than T We write

From the momentum equation (2.3), considering the leading order and noting that

2.1 Derivation of the vector Zakharov system 9

s4πe2N0

(2.9) with (2.10) implies that

−2iωc2e∂TE + ∇ × (∇ × E) − mγeTe

2

where, resulting from (2.17) and (2.19), the contribution of the ions is negligible

We rewrite the amplitude equation (2.21) as

2

2 e

2.1 Derivation of the vector Zakharov system 10where

of the mean electron field We thus replace (2.24) by

2.2 Simplification and generalization 11

This Zakharov system governs the coupled dynamics of the electric-field amplitudeand of the low-frequency density fluctuations of the ions and describes the dynamics

of the complex envelope of the electric field oscillations near the electron plasmafrequency and the slow variations of the density perturbations

In order to obtain a dimensionless form of the system (2.33)-(2.34), we define thenormalized variables

2

3ω2

eζ2 d

(2.38)and plugging (2.35)-(2.36) into (2.33)-(2.34), and then removing all primes, we getthe following dimensionless vector Zakharov system in three dimension

When we choose a = 1 in (2.39), the system (2.39)-(2.40) collapses to the standardvector Zakharov system

2.2 Simplification and generalization 12

into the system

(2.52)-(2.53) will be blowup, therefore in practice, if a linear damping term is added

to arrest blow up, one arrives at the generalized ZS (1.1)-(1.2)

2.3 Relation to the nonlinear Schr¨odinger equation(NLS) 13

For the generalized ZS (1.1)- (1.2), in the case of ε → 0 (corresponding to infinite

well-known nonlinear Schr¨odinger equation(NLS) without (γ = 0) or with (γ > 0) alinear damping term:

There are at least three conservation laws in the generalized ZS (1.1)-(1.2) withoutdamping (γ = 0) describing the propagation of Langmuir waves in plasma

Theorem 2.4.1 The generalized Zakharov system (ZS) (1.1)-(1.2) without ing term (γ = 0) preserves the conserved quantities They are the wave energy

damp-D =Z

the momentum

P =Z

R d

i

2.4 Conservation laws of the system 14and the Hamiltonian

Proof Multiplying (1.1) by E, the conjugate of E, we get

Then calculating the conjugate of (1.1) and multiplying it by E, one finds

Subtracting (2.61) from (2.60) and then multiplying both sides by −i, one gets

2.4 Conservation laws of the system 15

dx

2.5 Well-posedness of the Zakharov system (ZS) 16quantities collapse

Based on the conservation laws, C.Sulem and P.L.Sulem [40] prove the wellposednessfor the standard ZS (2.47)-(2.48)

2.6 Plane wave and soliton wave solutions 17

depends on the initial conditions

In one spatial dimension, the generalized ZS (1.1)- (1.2) collapses to

which admits plane wave and soliton wave solutions

Firstly, it is instructive to examine some explicit solutions to (2.74) and (2.75) Thewell-known plane wave solutions [29] can be given in the following form:

2γ (e −2γt −1)”

(b−a) 2, γ 6= 0,

(2.77)where r is an integer and c, d are constants

Secondly, as is well known, the standard ZS is not exactly integrable Therefore thegeneralized ZS cannot be exactly integrable, either However, it has exact one-solitonsolutions to (2.74) and (2.75) for γ = 0 [24, 25]:

Es(x, t; η, V, ε, ν) =

λ

2.6 Plane wave and soliton wave solutions 18

trivial phase constant

Finally, we will consider the periodic soliton solution with a period L in 1d of thestandard ZS, that is, d = 1, ε = 1, α = 1, λ = 0, γ = 0 and ν = −1 in (1.1)-(1.2).The analytic solution of the ZS (2.74)-(2.75) was derived in [26] and used to testdifferent numerical methods for the ZS in [31, 19, 12] The solution can be writtenas

Furthermore, we supplement (3.1)-(3.5) by imposing the compatibility condition

3.1 Time-splitting spectral discretizations (TSSP) 20

When γ = 0, D(t) ≡ D(0), i.e., it is an invariant of the ZS [12] Moreover, the ZSalso has the following properties

In some cases, the boundary conditions (3.4) and (3.5) may be replaced by

We choose the spatial mesh size h = ∆x > 0 with h = (b − a)/M for M being aneven positive integer, the time step being k = ∆t > 0 and let the grid points andthe time step be

discretized by Fourier spectral method in space and second-order central differencescheme in time, and equation (3.1) is solved in two splitting steps One solves first

for the time step of length k, and then

3.1 Time-splitting spectral discretizations (TSSP) 21

for the same time step Equation (3.10) will be discretized in space by the Fourier

by E, the conjugate of E, we get

Then calculating the conjugate of the ODE (3.11) and multiplying it by E, one finds

Subtracting (3.13) from (3.12) and then multiplying both sides by −i, one gets

E(x, tm+1) = e−iRtmtm+1[αN (x,τ )−λe −2γ(τ −tm) |E(x,t m )| 2 −iγ] dτ E(x, tm)

≈







e−ik[α(N (x,t m )+N (x,t m+1 ))/2−λ|E(x,t m )| 2 ]E(x, tm), γ = 0,

e−γk−i[kα(N (x,tm )+N (x,t m+1 ))/2+λ|E(x,t m )| 2 (e −2γk −1)/2γ] E(x, tm), γ 6= 0

3.1 Time-splitting spectral discretizations (TSSP) 22

where ( bU )l, the Fourier coefficient of a vector U = (U0, U1, U2, , UM)T with U0 =

This type of discretization for the initial condition (3.3) is equivalent to the use of the

to 0 exponentially fast as the mesh size h goes to 0

Note that the spatial discretization error of this method is of spectral-order accuracy

in h and time discretization error is of second-order accuracy in k, which will bedemonstrated in Chapter 4 by our numerical results

Note that a main advantage of the time-splitting spectral method is that if a constant

property also holds for the exact solution of the ZS, but does not hold for the finitedifference schemes proposed in [19, 12] and the spectral method proposed in [31], incontrast

3.1 Time-splitting spectral discretizations (TSSP) 23

Remark 3.1.1 If the periodic boundary conditions (3.4) and (3.5) are replaced

by (3.9), then the Fourier basis used in the above algorithm can be replaced bythe sine basis In fact, the generalized Zakharov system (3.1) and (3.2) with thehomogeneous periodic boundary condition (3.9) and initial condition (3.3) can bediscretized by

2M

Choose the initial data in (3.4)-(3.5) as

then 1d generalized ZS (2.74)-(2.75) admits the plane wave solution (2.76)-(2.77) Inthis case, our numerical method TSSP gives exact solution provided M ≥ 2(|r| + 1)

3.1 Time-splitting spectral discretizations (TSSP) 24Plugging (3.27)-(3.28) into (3.21)-(3.22), we get

3.1 Time-splitting spectral discretizations (TSSP) 25Here we use the identity

vuu

tb − aM

M −1X

j=0

vuu

tb − aM

kEmk2l 2 = e−2γtm

b − aM